\(\int \frac {x}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx\) [439]

Optimal result

Integrand size = 25, antiderivative size = 73 \[ \int \frac {x}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=-\frac {x}{b c (a+b \text {arcsinh}(c x))}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c^2}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c^2} \]

[Out]

-x/b/c/(a+b*arcsinh(c*x))+Chi((a+b*arcsinh(c*x))/b)*cosh(a/b)/b^2/c^2-Shi((a+b*arcsinh(c*x))/b)*sinh(a/b)/b^2/
c^2

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5818, 5774, 3384, 3379, 3382} \[ \int \frac {x}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c^2}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c^2}-\frac {x}{b c (a+b \text {arcsinh}(c x))} \]

[In]

Int[x/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2),x]

[Out]

-(x/(b*c*(a + b*ArcSinh[c*x]))) + (Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(b^2*c^2) - (Sinh[a/b]*Sinh
Integral[(a + b*ArcSinh[c*x])/b])/(b^2*c^2)

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 5774

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cosh[-a/b + x/b], x], x
, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 5818

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp
[((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] - Dist[f*(m/
(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]], Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x], x]
/; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {x}{b c (a+b \text {arcsinh}(c x))}+\frac {\int \frac {1}{a+b \text {arcsinh}(c x)} \, dx}{b c} \\ & = -\frac {x}{b c (a+b \text {arcsinh}(c x))}+\frac {\text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^2} \\ & = -\frac {x}{b c (a+b \text {arcsinh}(c x))}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^2}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^2} \\ & = -\frac {x}{b c (a+b \text {arcsinh}(c x))}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c^2}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\frac {-\frac {b c x}{a+b \text {arcsinh}(c x)}+\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b^2 c^2} \]

[In]

Integrate[x/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2),x]

[Out]

(-((b*c*x)/(a + b*ArcSinh[c*x])) + Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] - Sinh[a/b]*SinhIntegral[a/b + A
rcSinh[c*x]])/(b^2*c^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(150\) vs. \(2(73)=146\).

Time = 0.28 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.07

method	result	size
default	\(-\frac {-\sqrt {c^{2} x^{2}+1}+c x}{2 c^{2} b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right )}{2 c^{2} b^{2}}-\frac {\operatorname {arcsinh}\left (c x \right ) \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {a}{b}} b +\operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {a}{b}} a +b c x +\sqrt {c^{2} x^{2}+1}\, b}{2 c^{2} b^{2} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}\)	\(151\)

[In]

int(x/(a+b*arcsinh(c*x))^2/(c^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*(-(c^2*x^2+1)^(1/2)+c*x)/c^2/b/(a+b*arcsinh(c*x))-1/2/c^2/b^2*exp(a/b)*Ei(1,arcsinh(c*x)+a/b)-1/2/c^2/b^2
*(arcsinh(c*x)*Ei(1,-arcsinh(c*x)-a/b)*exp(-a/b)*b+Ei(1,-arcsinh(c*x)-a/b)*exp(-a/b)*a+b*c*x+(c^2*x^2+1)^(1/2)
*b)/(a+b*arcsinh(c*x))

Fricas [F]

\[ \int \frac {x}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {x}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x/(a+b*arcsinh(c*x))^2/(c^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(c^2*x^2 + 1)*x/(a^2*c^2*x^2 + (b^2*c^2*x^2 + b^2)*arcsinh(c*x)^2 + a^2 + 2*(a*b*c^2*x^2 + a*b)*a
rcsinh(c*x)), x)

Sympy [F]

\[ \int \frac {x}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \sqrt {c^{2} x^{2} + 1}}\, dx \]

[In]

integrate(x/(a+b*asinh(c*x))**2/(c**2*x**2+1)**(1/2),x)

[Out]

Integral(x/((a + b*asinh(c*x))**2*sqrt(c**2*x**2 + 1)), x)

Maxima [F]

\[ \int \frac {x}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {x}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x/(a+b*arcsinh(c*x))^2/(c^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

-(c^3*x^4 + c*x^2 + (c^2*x^3 + x)*sqrt(c^2*x^2 + 1))/((c^2*x^2 + 1)*a*b*c^2*x + ((c^2*x^2 + 1)*b^2*c^2*x + (b^
2*c^3*x^2 + b^2*c)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^3*x^2 + a*b*c)*sqrt(c^2*x^2 + 1))
+ integrate((c^5*x^5 + (c^2*x^2 + 1)*c^3*x^3 + 3*c^3*x^3 + 2*c*x + (2*c^4*x^4 + 3*c^2*x^2 + 1)*sqrt(c^2*x^2 +
1))/((c^2*x^2 + 1)^(3/2)*a*b*c^3*x^2 + 2*(a*b*c^4*x^3 + a*b*c^2*x)*(c^2*x^2 + 1) + ((c^2*x^2 + 1)^(3/2)*b^2*c^
3*x^2 + 2*(b^2*c^4*x^3 + b^2*c^2*x)*(c^2*x^2 + 1) + (b^2*c^5*x^4 + 2*b^2*c^3*x^2 + b^2*c)*sqrt(c^2*x^2 + 1))*l
og(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^5*x^4 + 2*a*b*c^3*x^2 + a*b*c)*sqrt(c^2*x^2 + 1)), x)

Giac [F]

\[ \int \frac {x}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {x}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x/(a+b*arcsinh(c*x))^2/(c^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(x/(sqrt(c^2*x^2 + 1)*(b*arcsinh(c*x) + a)^2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {c^2\,x^2+1}} \,d x \]

[In]

int(x/((a + b*asinh(c*x))^2*(c^2*x^2 + 1)^(1/2)),x)

[Out]

int(x/((a + b*asinh(c*x))^2*(c^2*x^2 + 1)^(1/2)), x)